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Mathematics > Geometric Topology

Title:
Floer homology and knot complements

Abstract: We use the Ozsvath-Szabo theory of Floer homology to define an invariant of
knot complements in three-manifolds. This invariant takes the form of a
filtered chain complex, which we call CF_r. It carries information about the
Floer homology of large integral surgeries on the knot. Using the exact
triangle, we derive information about other surgeries on knots, and about the
maps on Floer homology induced by certain surgery cobordisms. We define a
certain class of \em{perfect} knots in S^3 for which CF_r has a particularly
simple form. For these knots, formal properties of the Ozsvath-Szabo theory
enable us to make a complete calculation of the Floer homology. This is the
author's thesis; many of the results have been independently discovered by
Ozsvath and Szabo in math.GT/0209056.